Give me some difficult exponentand root form practice

Here are some challenging exponent and root form practice problems:

### Problem Set

1. **Simplify the expression:**
   \[
   \left(\frac{16x^8y^{12}}{4x^2y^6}\right)^{\frac{3}{4}}
   \]

2. **Solve for \(x\):**
   \[
   \sqrt[3]{x^4 \cdot \sqrt{x^5}} = x^3 \cdot \sqrt[4]{x}
   \]

3. **Simplify the expression:**
   \[
   (2^{2/3} \cdot 4^{3/4}) \div (8^{1/3} \cdot 16^{1/2})
   \]

4. **Rewrite the expression in radical form and simplify:**
   \[
   (x^{5/6} \cdot y^{1/3})^2 \cdot \left(\frac{1}{z^{1/4}}\right)^2
   \]

5. **Evaluate the following expression:**
   \[
   \left(\frac{9}{16}\right)^{3/2} \cdot \left( \frac{81}{256} \right)^{1/4}
   \]

### Tip:
When dealing with exponents and roots, remember to carefully apply the laws of exponents, such as \(a^{m/n} = \sqrt[n]{a^m}\), and always simplify expressions where possible before performing more complex operations.

Would you like solutions to any of these problems or have any questions?

### Related Questions:
1. How do you simplify expressions with negative exponents and fractional bases?
2. What is the general method for solving equations involving radicals and exponents?
3. How can exponents be applied when dealing with polynomial division?
4. What are the key differences between fractional exponents and roots?
5. How do you convert a complex expression with multiple exponents into a simplified form?



Explore challenging exponent and root practice problems with detailed solutions. Includes problems on simplifying expressions, solving equations involving exponents and roots, and evaluating complex expressions. Suitable for high school students studying advanced algebra and exponent rules.

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